### Nuprl Lemma : unsquashed-monotone-bar-induction8-false3

`¬(∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.`
`    ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. Q[n + 1;s.m@n]) `` Q[n;s]))`
`    `` (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. ∀m:{n...}. B[m;f]))`
`    `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` Q[n;s]))`
`    `` Q[0;λx.⊥]))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` seq-add: `s.x@n` int_upper: `{i...}` int_seg: `{i..j-}` nat: `ℕ` bottom: `⊥` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` true: `True` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` nat: `ℕ` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` so_apply: `x[s]` int_upper: `{i...}` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` exists: `∃x:A. B[x]` so_lambda: `λ2x y.t[x; y]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k`
Lemmas referenced :  int_formula_prop_less_lemma intformless_wf int_seg_properties seq-add_wf le_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties equiv_rel_true true_wf subtype_rel_self false_wf int_seg_subtype_nat int_seg_wf subtype_rel_dep_function int_upper_subtype_nat int_upper_wf exists_wf quotient_wf nat_wf all_wf unsquashed-monotone-bar-induction8-false
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution independent_functionElimination thin isectElimination functionEquality hypothesis because_Cache sqequalRule lambdaEquality setElimination rename hypothesisEquality applyEquality natural_numberEquality independent_isectElimination independent_pairFormation dependent_set_memberEquality addEquality dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll cumulativity universeEquality instantiate productElimination

Latex:
\mneg{}(\mforall{}B,Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  Q[n  +  1;s.m@n])  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  B[m;f]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  Q[0;\mlambda{}x.\mbot{}]))

Date html generated: 2016_05_14-PM-09_45_35
Last ObjectModification: 2016_02_02-PM-04_40_10

Theory : continuity

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